Subgroups — C₁₇×#2943 #14525949

Order 2176.

492 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₄×#41×C₁₇	C₂	1	2	yes	maximal, complemented
3	C₁₇×#442	C₂	1	2	yes	maximal, complemented
4	C₁₇×#441	C₂	1	2	yes	maximal, complemented
5	C₁₇×#440	C₂	1	4	yes	maximal, complemented
6	C₁₇×#427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#416	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₇	C₂²	1	2	yes	—
10	C₁₇×#135	C₄	1	8	yes	complemented
11	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₁₇	C₂²	1	2	yes	—
13	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₁₇	C₂²	1	4	yes	—
17	C₄×D₄×C₁₇	—	2	4	no	—
18	C₄×D₄×C₁₇	C₂²	1	4	yes	—
19	C₁₇×#117	—	2	2	no	—
20	C₁₇×#117	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₇	—	2	2	no	—
22	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
23	C₁₇×#107	C₂²	1	2	yes	—
24	C₁₇×#106	C₂²	1	2	yes	—
25	C₁₇×#103	C₂²	1	4	yes	—
26	C₁₇×#102	C₂²	1	4	yes	—
27	C₁₇×#97	C₂²	1	2	yes	—
28	C₁₇×#97	C₂²	1	2	yes	—
29	#41×C₁₇	—	2	4	no	—
30	#41×C₁₇	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₁₇	C₂³	1	2	yes	—
32	C₂×D₄×C₁₇	—	2	2	no	—
33	C₂×D₄×C₁₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₇	—	2	2	no	—
35	C₂²×C₄×C₁₇	—	2	4	no	—
36	C₂²×C₄×C₁₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
38	Q₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
39	SD₁₆×C₁₇	C₂×C₄	1	8	yes	complemented
40	D₈×C₁₇	C₂×C₄	1	4	yes	complemented
41	C₂×C₈×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	C₂×C₄	1	4	yes	complemented
43	C₂×C₈×C₁₇	C₂³	1	2	yes	—
44	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₁₇	—	2	4	no	—
47	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₁₇	—	2	2	no	—
49	C₂²⋊C₄×C₁₇	—	2	4	no	—
50	C₄²×C₁₇	—	2	4	no	—
51	C₄²×C₁₇	—	2	2	no	—
52	C₄²×C₁₇	D₄	1	2	yes	—
53	C₄²×C₁₇	C₂³	1	2	yes	—
54	C₂³×C₁₇	—	2	2	no	—
55	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₁₇	—	2	2	no	—
57	Q₈×C₁₇	C₂²×C₄	1	4	yes	—
58	D₄×C₁₇	—	2	8	no	—
59	D₄×C₁₇	—	2	2	no	—
60	D₄×C₁₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	8	no	—
63	C₂×C₄×C₁₇	—	2	4	no	—
64	C₂×C₄×C₁₇	—	4	2	no	—
65	C₂×C₄×C₁₇	—	4	8	no	—
66	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
68	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	#41	1	2	yes	—
70	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
71	C₈×C₁₇	—	2	2	no	—
72	C₈×C₁₇	C₂²×C₄	1	2	yes	—
73	C₈×C₁₇	C₂²×C₄	1	2	yes	—
74	#2943	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₁₇	—	2	2	no	—
76	C₂²×C₁₇	—	2	4	no	—
77	C₂²×C₁₇	—	4	4	no	—
78	C₂²×C₁₇	C₄×D₄	1	2	yes	—
79	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₁₇	—	2	4	no	—
81	C₄×C₁₇	—	2	4	no	—
82	C₄×C₁₇	—	4	4	no	—
83	C₄×C₁₇	C₄×D₄	1	2	yes	—
84	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
86	C₂×#135	C₂×C₁₇	1	1	yes	characteristic
87	C₄×#41	C₂×C₁₇	1	2	yes	complemented
88	#442	C₂×C₁₇	1	2	yes	complemented
89	#441	C₂×C₁₇	1	2	yes	complemented
90	#440	C₂×C₁₇	1	4	yes	complemented
91	#427	C₂×C₁₇	1	1	yes	characteristic, complemented
92	#416	C₂×C₁₇	1	2	yes	complemented
93	C₂×#97	C₂×C₁₇	1	1	yes	characteristic, complemented
94	C₂×C₁₇	—	2	2	no	—
95	C₂×C₁₇	—	4	4	no	—
96	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂×C₁₇	#575	1	2	yes	—
98	C₂×#41	C₂²×C₁₇	1	2	yes	—
99	#135	C₄×C₁₇	1	8	yes	complemented
100	C₂×Q₁₆	C₂²×C₁₇	1	1	yes	characteristic
101	C₂×SD₁₆	C₂²×C₁₇	1	2	yes	—
102	C₂×D₈	C₂²×C₁₇	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
105	C₄×Q₈	C₂²×C₁₇	1	4	yes	—
106	C₄×D₄	—	2	4	no	—
107	C₄×D₄	C₂²×C₁₇	1	4	yes	—
108	#117	—	2	2	no	—
109	#117	C₂²×C₁₇	1	2	yes	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
112	#107	C₂²×C₁₇	1	2	yes	—
113	#106	C₂²×C₁₇	1	2	yes	—
114	#103	C₂²×C₁₇	1	4	yes	—
115	#102	C₂²×C₁₇	1	4	yes	—
116	#97	C₂²×C₁₇	1	2	yes	—
117	#97	C₂²×C₁₇	1	2	yes	—
118	C₁₇	#2943	1	1	yes	characteristic, direct product factor, Sylow
119	#41	—	2	4	no	—
120	#41	C₂×C₄×C₁₇	1	8	yes	complemented
121	C₂×Q₈	C₂³×C₁₇	1	2	yes	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	C₂³×C₁₇	1	2	yes	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	D₄×C₁₇	1	2	yes	—
127	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
128	Q₁₆	C₂×C₄×C₁₇	1	4	yes	complemented
129	SD₁₆	C₂×C₄×C₁₇	1	8	yes	complemented
130	D₈	C₂×C₄×C₁₇	1	4	yes	complemented
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	C₂×C₄×C₁₇	1	4	yes	complemented
133	C₂×C₈	C₂³×C₁₇	1	2	yes	—
134	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
135	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	4	no	—
142	C₄²	D₄×C₁₇	1	2	yes	—
143	C₄²	C₂³×C₁₇	1	2	yes	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂×D₄×C₁₇	1	1	yes	characteristic
146	Q₈	—	2	2	no	—
147	Q₈	C₂²×C₄×C₁₇	1	4	yes	—
148	D₄	—	2	8	no	—
149	D₄	—	2	2	no	—
150	D₄	C₂²×C₄×C₁₇	1	4	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	8	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	8	no	—
156	C₂×C₄	C₂²×C₄×C₁₇	1	2	yes	—
157	C₂×C₄	C₂×D₄×C₁₇	1	4	yes	—
158	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
159	C₂×C₄	#41×C₁₇	1	2	yes	—
160	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	C₂²×C₄×C₁₇	1	2	yes	—
163	C₈	C₂²×C₄×C₁₇	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	4	4	no	—
167	C₂²	C₄×D₄×C₁₇	1	2	yes	—
168	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	2	4	no	—
171	C₄	—	4	4	no	—
172	C₄	C₄×D₄×C₁₇	1	2	yes	—
173	C₄	C₂³×C₄×C₁₇	1	1	yes	characteristic
174	C₄	C₂×#41×C₁₇	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	4	4	no	—
177	C₂	C₂×C₄×D₄×C₁₇	1	1	yes	characteristic
178	C₂	C₁₇×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂×D₄
commutator	C₄	C₂³×C₄×C₁₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×C₁₇×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄×#41×C₁₇	C₂	semidirect
3. C₁₇×#442	C₂	semidirect
4. C₁₇×#441	C₂	semidirect
5. C₁₇×#440	C₂	semidirect
6. C₁₇×#427	C₂	semidirect
7. C₁₇×#416	C₂	semidirect
8. C₂×C₁₇×#97	C₂	semidirect
10. C₁₇×#135	C₄	semidirect
30. #41×C₁₇	C₂×C₄	semidirect
38. Q₁₆×C₁₇	C₂×C₄	semidirect
39. SD₁₆×C₁₇	C₂×C₄	semidirect
40. D₈×C₁₇	C₂×C₄	semidirect
42. C₂×C₈×C₁₇	C₂×C₄	semidirect
74. #2943	C₁₇	direct (N × H)
87. C₄×#41	C₂×C₁₇	semidirect
88. #442	C₂×C₁₇	semidirect
89. #441	C₂×C₁₇	semidirect
90. #440	C₂×C₁₇	semidirect
91. #427	C₂×C₁₇	semidirect
92. #416	C₂×C₁₇	semidirect
93. C₂×#97	C₂×C₁₇	semidirect
99. #135	C₄×C₁₇	semidirect
118. C₁₇	#2943	direct (N × H)
120. #41	C₂×C₄×C₁₇	semidirect
128. Q₁₆	C₂×C₄×C₁₇	semidirect
129. SD₁₆	C₂×C₄×C₁₇	semidirect
130. D₈	C₂×C₄×C₁₇	semidirect
132. C₂×C₈	C₂×C₄×C₁₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₇×#135	2
C₄×#41×C₁₇	2
C₁₇×#442	2
C₁₇×#441	2
C₁₇×#440	2
C₁₇×#427	2
C₁₇×#416	2
C₂×C₁₇×#97	2
#2943	17

Subgroups — C17×#2943 #14525949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2943 #14525949